Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate for negative integer values of n.
Throughout this section, we assume that with and .
By substituting (1) into (2), we obtain the following functional equation:
By using this functional equation, we arrive at the following theorem.
Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have
By using (5), we modify (3) as follows:
By using (7), we derive the following result.
Corollary 2.2 Let χ be a Dirichlet character of conductor . Then we have
By applying the Mellin transformation to the generating function (1), Ozden et al. [, p.2784 Equation (4.1)] gave an integral representation of the unified zeta function :
where the additional constraint is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [, p.2784 Equation (4.1)] defined the unified zeta function as follows:
By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions :
in terms of the generating function defined in (7). By substituting (9) into ( 11), we obtain
where ( (); ()).
Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions as follows.
Definition 2.3 Let χ be a Dirichlet character of conductor . For (), we define a two-variable unified L-function by
Remark 2.4 If we substitute into (13), we get the unified L-function
where (, ()).
Remark 2.5 Upon substituting and into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:
where, for a positive integer r, ξ is the r th root of 1.
Remark 2.6 Substituting into (13), we get a unification of the L-functions
Substituting into (13), we get a unification of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function
(cf. [27, 28]) and the Riemann zeta function
are obvious special cases of the unified zeta function (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function was given by Ozden et al. :
where the Hurwitz-Lerch zeta function is defined by
which converges for (, when ; when ), where as usual
(cf. [27, 28]).
A relationship between the functions and is provided by the next theorem.
Theorem 2.7 Let . Let χ be a Dirichlet character of conductor . Then we have
Proof Substituting , , into (13), we obtain
After some algebraic manipulations, we arrive at the desired result. □
Remark 2.8 Substituting into (13), we have
which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:
Let f be an odd integer. If we set and into (13), then we have
which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:
By using (15) and (14), we arrive at the following result.
Corollary 2.9 Let . Let χ be a Dirichlet character of conductor . Then we have
Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have
Proof By substituting into (15), we get
By using Theorem 7 in , we get
By substituting (8) into the above, we arrive at the desired result. □
Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions defined by (13). We thus have (cf. )
where . By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have
where and , the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that is non-zero at . Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).